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  "body": "\n## Summary\n\nDeterminacy is the property of an infinite two-player game of perfect information that one of the two players has a winning strategy.\nIn the games of descriptive set theory, the players alternately choose natural numbers forever.\nThe first player wins when the resulting [[sequence|0G9EKM0E15]] lies in a fixed payoff set.\nGale and Stewart proved that games with an open or a closed payoff set are determined.\nThe [[Borel determinacy theorem|8FVWYEZWTK]] extends this to all Borel payoff sets.\nStronger determinacy hypotheses, such as [[projective determinacy|2GY3131ZPS]] and the [[axiom of determinacy|PEXTM3MJ5Z]], imply strong regularity properties of sets of reals.\n\n## Explanations\n\n### L1 human\n\nTwo players take turns writing down numbers, and they never stop.\nBefore the game starts, a rule decides which infinite sequences count as a win for the first player.\nA strategy is a recipe that tells a player which number to write in each position, and it is winning if it wins against every way the opponent can play.\nIn a finite game where every play ends with a win for one of the players, one of them always has a winning strategy.\nFor infinite games this is no longer automatic, and determinacy is the statement that some player has a winning strategy.\n",
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